We're going to start our discussion of number theory with two familiar topics: odd versus even numbers and rational numbers. Our main focus is going to be on learning different methods that can be used to prove mathematical theorems. You'll also see how formal definitions give us the precision that we need to make proofs happen.
Video Lectures
We’re going to be approaching our introduction to number theory from two different directions. On the one hand, we're going to use it as a vehicle for introducing basic methods of proving mathematical statements and, on the other hand, we're going to see how it can be applied to practical fields like cryptography and algorithm design. In this lecture, we're going to talk about what constitutes a proof and look at an example of how the logical forms we've talked about previously justify the steps we take. (lecture slides)
We're going to start our discussion of proofs and number theory with a discussion of mathematical definitions. They're the cornerstone that practically all mathematics is based on and they've got a very specific format and it's important to understand the information that it's giving you. (lecture slides)
We're going to start our number theory with something you're already familiar with: odd numbers versus even numbers. In this lecture, we're going to introduce a formal definition that is consistent with what you already know but is strong enough to let us prove things with it. (lecture slides)
Now that we have a formal, practical definition of odd and even numbers, we're ready to start thinking about techniques that you can use to prove mathematical statements. (lecture slides)
Rational numbers aren't really a part of number theory but they're made up of integers which are and they're another good source of basic theorems for us to practice proving. (lecture slides)
